There are a couple of important takeaways from this analysis. The first is simply that the phase \(\Phi \)  must be multi-valued, even in the context of relativistic fields.Although simple scalar fields such as this are, of course, not found in nature, the fact that this property holds even in this relatively stunted model seems to imply that we should certainly expect this structure to persist with other Bosonic fields, and certainly for Fermion fields, as well. Simply put, single-valued phases simply do not recover states of physical importance. This has some important implications for the ED approach. For instance, in previous iterations of ED (see e.g., [1]}[2]}) it was posited that the drift potential \(\phi \)  could possibly be an entropy. Clearly this explanation is not sufficient since entropy must be single-valued. Instead the drift potential and its attendant angular properties must have a deeper geometrical interpretation. This is made rather manifest in [3]} wherein the geometry of the e-phase space is assumed to be spherical. Thus if probabilities in the e-phase space play the role of a “polar" coordinate, then it is natural that the drift potential satisfies the role of its corresponding phase angle.
